When does propagation of chaos in the critical Curie-Weiss model stop?

Abstract

We study increasing propagation of chaos for the critical Curie- Weiss model (i.e. the mean-field Ising model at critical inverse temperature 1, with no external field). We give a simple way to see that for windows of size k(N) of smaller order than sqrt{N} we still have propagation of chaos (reproving earlier results, see e.g.[1]), while for k(N) of order sqrt{N} the propagation of chaos breaks down. The law of a single spin converges to π, the Bernoulli law with parameter 1/2. If k(N) = alpha sqrt{N}, we give an explicit formula for the limiting distance in total variation of the law of the first k(N) spins with respect to the k-fold product of pi, as a function of alpha. For even larger window sizes, the distribution of the spins has, in the thermodynamical limit, maximal distance to the k-fold product of pi. One of the ingredients of the proof is a result about the unimodality/non- unimodality of the law of the number of positive spins among the first k = k(N) spins, which may be of independent interest.